Meaning "execute in the electric chair" is U.S. slang from 1929. To go out of the frying pan into the fire is first attested in Thomas More (1532). The related noun is from 1630s. Related: Fried; frying. Frying pan recorded from mid-14c.

fry

n.

"young fish," late 13c., from Anglo-French frei, from Old French frai "spawn," from froier "to rub, spawn (by rubbing abdomen on sand)." First applied to human offspring 14c. in Scottish, though OED and some other sources trace this usage to Old Norse frjo, fræ "seed, offspring."